Of course! Let's break down both parts of the question and explain how to solve them:
A. Find the number z such that the proportion of observations that are less than z in a standard Normal distribution is 0.8.
To solve this, we need to find the z-score corresponding to a cumulative probability of 0.8 in a standard Normal distribution. The z-score represents the number of standard deviations an observation is away from the mean.
To find the answer, you can use a standard Normal distribution table (also known as a z-table) or a calculator with built-in statistical functions. These tools will allow you to look up or calculate the z-value for a given probability.
1. Using a z-table: Locate the value closest to 0.8 in the body of the table. The row headers represent the tenths digit, and the column headers represent the hundredths digit. The intersection of these values will give you the z-score.
2. Using a calculator: Many calculators or statistical software have functions to find the z-score directly. You can input the probability of 0.8 and obtain the corresponding z-value.
Once you have the z-score, you can apply it to the mean and standard deviation of the standard Normal distribution to find the corresponding value, z.
B. Find the number z such that 35% of all observations from a standard Normal distribution are greater than z.
Similar to part A, we need to find the z-score corresponding to a cumulative probability. However, this time we are interested in the proportion of observations that are greater than z, rather than less than z.
To solve this, you can follow the same steps as in part A, but with a slight modification. Instead of looking up the probability in the z-table or calculator, you need to find the probability of the area to the left of z. Subtract this probability from 1, and you will obtain the proportion of observations greater than z. Next, equate this proportion to 0.35 (since we want 35% of observations to be greater than z) and solve for z using the z-table or calculator.
By applying these steps, you should be able to find the answers to both parts of the question.